Problem: Find $\dfrac{d}{dx}[9\log_2(x)]$. Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{9\ln(x)}{\ln(2)}$ (Choice B) B $\dfrac{9}{x\ln(2)}$ (Choice C) C $\dfrac{9\log_2(x)}{x}$ (Choice D) D $\dfrac{9}{x\log_2(x)}$
Solution: The expression to differentiate includes a logarithmic term. Remember that the derivative of the general logarithmic term $\log_a(x)$ (where $a$ is any positive constant and $a\neq 1$ ) is $\dfrac{1}{\ln(a)\cdot x}$. Put another way, $\dfrac{d}{dx}[\log_a(x)]=\dfrac{1}{\ln(a)\cdot x}$. [Is there an easy way to memorize that?] We can use this to find the derivative as shown below. $\begin{aligned} &\phantom{=}\dfrac{d}{dx}[9\log_2(x)] \\\\ &=9\dfrac{d}{dx}[\log_2(x)] \\\\ &=9\cdot\dfrac{1}{\ln(2)x} \\\\ &=\dfrac{9}{x\ln(2)} \end{aligned}$ In conclusion, $\dfrac{d}{dx}[9\log_2(x)]=\dfrac{9}{x\ln(2)}$.